MATH 2270-2
Final Exam Sample Problems
!
1
1. (5 points) Let A be a 2 × 2 matrix such that A
=
1
Spring 2016
!
!
1
2
. Compute A
.
0
2
2. (5 points) State (1) the definition of norm, (2) the Cauchy-Schwartz inequality and
(3) the triangle inequality, for vectors in Rn .
3. (5 points) Suppose A = B(C + D)E and all the matrices are n × n invertible. Find an
equation for C.
4. (5 points) Find all solutions to the system of equations
2w + 3x + 4y + 5z = 1
4w + 3x + 8y + 5z = 2
6w + 3x + 8y + 5z = 1
5. (5 points) Let A =
2 1
0 3
!
. Show the details of two different methods for finding
the inverse of the matrix A.
6. (5 points) 
Find
2

the matrix A =  1
0
a
1
2
1
factorization
A = LU into lower and upper triangular matrices for

0

1 .
2
7. (5 points) Let Q be a 2 × 2 matrix with QQT = I. Prove that Q has columns of unit
length and its two columns are orthogonal.
8. (5 points) True or False? If the 3 × 3 matrices A and B are triangular, then AB is
triangular.
9. (5 points) True or False? If a 3 × 3 matrix A has an inverse, then for all vectors ~b the
equation A~x = ~b has a unique solution ~x.
10. (5 points) Let A be a 3 × 4 matrix. Find the elimination matrix E which under left
multiplication against A performs both (1) and (2) with one matrix multiply.
1
(1) Replace Row 2 of A with Row 2 minus Row 1.
(2) Replace Row 3 of A by Row 3 minus 5 times Row 2.
11. (10 points) Determinant problem, chapter 3.
(a) [10%] True or False? The value of a determinant is the product of the diagonal elements.
(b) [10%] True or False? The determinant of the negative of the n × n identity matrix is −1.
(c) [30%] Assume given 3 × 3 matrices A, B. Suppose E2 E1 A2 = AB and E1 , E2 are
elementary matrices representing respectively a combination and a multiply by 3. Assume
det(B) = 27. Let C = −A. Find all possible values of det(C).
−1
(d) [20%] Determine
 all values ofx for which (2I + C) fails to exist, where I is the 3 × 3
2 x −1


identity and C =  3x 0 1 .
1 0 −1
(e) [30%] Let symbols a, b, c denote constants and define



A=

1 −1 0 0
1
0 0 0
a
b 0 1
1
c 1 12





Apply the adjugate [adjoint] formula for the inverse
A−1 =
adj(A)
|A|
to find the value of the entry in row 4, column 2 of A−1 .
12. (5 points) Define matrix A,

−2
3

A =  0 −4
1
4
vector ~b and vector variable ~x by the equations





0
−3
x1





0  , ~b =  5  , ~x =  x2  .
1
1
x3
For the system A~x = ~b, find x3 by Cramer’s Rule, showing all details (details count 75%).
To save time, do not compute x1 , x2 !


3 1 0


13. (5 points) Define matrix A =  3 3 1 . Find a lower triangular matrix L and
0 2 4
an upper triangular matrix U such that A = LU .
2
14. (5 points) Determine which values of k correspond
system A~x = ~b given by



1
4
k



A =  0 k − 2 k − 3  , ~b = 
1
4
3
to a unique solution for the

1

−1  .
k
15. (10 points) Let a, b and c denote constants and consider the system of equations

  

1
−b
c
x
a

  

c
a   y  =  −a 
 1
2 −b + c −a
z
−a
Use techniques learned in this course to briefly explain the following facts. Only write what
is needed to justify a statement.
(a). The system has a unique solution for (b + c)(2a + c) 6= 0.
(b). The system has no solution if 2a+c = 0 and a 6= 0 (don’t explain the other possibilities).
(c). The system has infinitely many solutions if a = c = 0 (don’t explain the other possibilities).
16. (5 points) Explain how the span theorem applies to show that the set S of all
linear combinations of the functions cosh x, sinh x is a subspace of the vector space V of all
continuous functions on −∞ < x < ∞.
17. (5 points) Write a proof that the subset S of all solutions ~x in Rn to a homogeneous
matrix equation A~x = ~0 is a subspace of Rn . This is called the kernel theorem.
18. (5 points) Using the subspace criterion, write two hypotheses that imply that a set
S in a vector space V is not a subspace of V . The full statement of three such hypotheses
is called the Not a Subspace Theorem.


0 1 1


19. (5 points) Report which columns of A are pivot columns: A =  0 1 2 .
0 0 0
3
20. (5 points) Find the complete solution ~x = ~xh + ~xp for the nonhomogeneous system


  
0 1 1
x1
2


  
 0 1 2   x2  =  3  .
0 0 0
x3
0
The homogeneous solution ~xh is a linear combination of Strang’s special solutions. Symbol
~xp denotes a particular solution.
21. (5 points) Find the vector general

1 0

A = 3 0
4 0
solution ~x to the equation A~x = ~b for

 
0 4
0
 ~  
1 0 , b = 4
0 1
0

0 1 1


22. (5 points) Find the reduced row echelon form of the matrix A =  0 0 0 .
0 1 2

23. (5 points) A 10 × 13 matrix A is given and the homogeneous system A~x = ~0
is transformed to reduced row echelon form. There are 7 lead variables. How many free
variables?
24. (5 points) The rank of a 10 × 13 matrix A is 7. Find the nullity of A.
!
!
!
3
4
10
25. (5 points) Given a basis ~v1 =
, ~v2 =
of R2 , and ~v =
, then ~v =
2
4
4
c1~v1 + c2~v2 for a unique set of coefficients c1 , c2 , called the coordinates of ~v relative to the
basis ~v1 , ~v2 . Compute c1 and c2 .
26. (5 points) Determine independence or dependence for the list of vectors
     
1
4
3
     
2 , 0 , 2
3
4
1
27. (5 points) Check the independence tests which apply to prove that 1, x2 , x3 are
independent in the vector space V of all functions on −∞ < x < ∞.
4
Wronskian test
Rank test
Determinant test
Euler Atom test
Wronskian of f~1 , f~2 , f~3 nonzero at x = x0 implies independence of f~1 , f~2 , f~3 .
Vectors ~v1 , ~v2 , ~v3 are independent if their augmented
matrix has rank 3.
Vectors ~v1 , ~v2 , ~v3 are independent if their square augmented matrix has nonzero determinant.
Any finite set of distinct atoms is independent.
Sample test
Functions f~1 , f~2 , f~3 are independent if a sampling matrix
has nonzero determinant.
Pivot test
Vectors ~v1 , ~v2 , ~v3 are independent if their augmented
matrix A has 3 pivot columns.
Orthogonality test A set of nonzero pairwise orthogonal vectors is independent.
28. (5 points) Define S to be the set of all vectors ~x in R3 such that x1 + x3 = 0 and
x3 + x2 = x1 . Prove that S is a subspace of R3 .
29. (5 points) The 5 × 6 matrix A below has some independent columns. Report the
independent columns of A, according to the Pivot Theorem.


0 0 0
0 0
0


 −3 0 0 −2 1 −1 



A=
−1
0
0
0
1
0




6 0
3 
 6 0 0
2 0 0
2 0
1
30. (5 points) Let S be the subspace of R4 spanned by the vectors
 
1
1
 
~v1 =   ,
1
0
 
1
1
 
~v2 =   .
0
1
Find a Gram-Schmidt orthonormal basis of S.
31. (5 points) Find the orthogonal projection vector ~v (the shadow projection vector) of
5
~v2 onto ~v1 , given



~v1 = 

1
1
1
0




,



~v2 = 

1
1
0
1



.

32. (5 points) Let A be an m × n matrix with independent columns. Prove that AT A is
invertible.
33. (5 points) Let A be an m × n matrix with AT A invertible. Prove that the columns
of A are independent.
34. (5 points) Let A be an m × n matrix and ~v a vector orthogonal to the nullspace of
A. Prove that ~v must be in the row space of A.
35. (5 points) Define matrix A and vector ~b by the equations
 


1
−2
3
0
 ~  

A =  0 −2
4 , b =  2 .
3
1
0 −2
Find the value of x2 by Cramer’s Rule in the system A~x = ~b.
!
2 −6
36. (5 points) Assume A−1 =
. Find the inverse of the transpose of A.
0
4
37. (5 points) This problem uses the identity A adj(A) = adj(A)A = |A|I, where |A| is the
determinant of matrix A. Symbol adj(A) is the adjugate or adjoint of A. The identity is used to derive
the adjugate inverse identity A−1 = adj(A)/|A|..
Let B be the matrix given below, where ? means the value of the entry does not affect
the answer to this problem. The second matrix is C = adj(B). Report the value of the
determinant of matrix C −1 B 2 .




1 −1 ? ?
4 4
2 0
 1
 −4 4 −2 0 
? 0 0 




B=
, C = 

 ?
 0 0
0 2 ? 
4 0 
?
0 0 ?
0 0
0 4
38. (5 points) Display the entry in row 3, column 4 of the adjugate matrix [or adjoint
6



matrix] of A = 

0
0
1
0

2 −1 0
0
4 1 

. Report both the symbolic formula and the numerical
3 −2 0 
1
1 0
value.
39. (5 points) Consider a 3 × 3 real matrix A with eigenpairs



  



5
i
−i



  



−1,  6  , 2i,  2  , −2i,  2  .
−4
0
0
Display an invertible matrix P and a diagonal matrix D such that AP = P D.



40. (5 points) Find the eigenvalues of the matrix A = 

0 −12
3 0
0
1 −1 0
0
1
3 0
0
5
1 3



.

To save time, do not find eigenvectors!


0 −12
3


41. (5 points) The matrix A =  0
1 −1  has eigenvalues 0, 2, 2 but it is not
0
1
3
diagonalizable, because λ = 2 has only one eigenpair. Find an eigenvector for λ = 2.
To save time, don’t find the eigenvector for λ = 0.
42. (5 points) Find the two complex eigenvectors
corresponding to complex eigenvalues
!
−1
2
−1 ± 2i for the 2 × 2 matrix A =
.
−2 −1
43. (5 points) Let A =
1,
−7 4
−12 7
1
2
!
. Circle possible eigenpairs of A.
!!
,
2,
2
1
!!
,
−1,
2
3
!!
.
44. (5 points) Let I denote the 3 × 3 identity matrix. Assume given two 3 × 3 matrices
B, C, which satisfy CP = P B for some invertible matrix P . Let C have eigenvalues −1, 1,
5. Find the eigenvalues of A = 2I + 3B.
7
45. (5 points) Let A be a 3 × 3 matrix with eigenpairs
(4, ~v1 ),
(3, ~v2 ),
(1, ~v3 ).
Let P denote the augmented matrix of the eigenvectors ~v2 , ~v3 , ~v1 , in exactly that order.
Display the answer for P −1 AP . Justify the answer with a sentence.
46. (5 points) The matrix A below has eigenvalues 3, 3 and 3. Test A to see it is
diagonalizable, and if it is, then display three eigenpairs of A.


4 1 1


A =  −1 2 1 
0 0 3
47. (5 points) Assume A is a given 4 × 4 matrix with eigenvalues 0, 1, 3 ± 2i. Find the
eigenvalues of 4A − 3I, where I is the identity matrix.


0 −2 −5
0 0


 3
0 −12
3 0 


48. (5 points) Find the eigenvalues of the matrix A = 
0
1 −1 0 
.
 0


0
1
3 0 
 0
0
0
5
1 3
To save time, do not find eigenvectors!
49. (5 points) Consider a 3 × 3 real matrix A with
  

 
i
13
  

 
6  , 2i,  2  ,
3, 
0
−41
eigenpairs



−i



−2i,  2  .
0
(1) [10%] Display an invertible matrix P and a diagonal matrix D such that
AP = P D.
(2) [10%] Display a matrix product formula for A, but do not evaluate the matrix
products, in order to save time.
50. (5 points) Assume two 3 × 3 matrices A, B have exactly the same characteristic
equations. Let A have eigenvalues 2, 3, 4. Find the eigenvalues of (1/3)B − 2I, where I is
the identity matrix.
8
51. (5 points) Let 3 × 3 matrices A and B be related by AP = P B for some invertible
matrix P . Prove that the roots of the characteristic equations of A and B are identical.
52. (5 points) Find the eigenvalues of the

2
 0

B=
 0
0
matrix B:

4 −1 0
5 −2 1 


0
4 1 
0
1 4




1 1
1




53. (5 points) Let W be the column space of A =  1 1  and let ~b =  −1 . Let
1 0
1
~
~
b̂ be the near point to ~b in the subspace W . Find b̂.
54. (5 points) There are real 2 × 2 matrices A such that A2 = −4I, where I is the identity
matrix. Give an example of one such matrix A and then verify that A2 + 4I = 0.
55. (5 points) Let Q =< ~q1 |~q2 > be orthogonal 2×2 and D a diagonal matrix with diagonal
entries λ1 , λ2 . Prove that the 2 × 2 matrix A = QDQT satisfies A = λ1 ~q1 ~q1T + λ2 ~q2 ~q2T .
56. (5 points) A matrix A is defined to be positive definite if and only if ~xT A~x > 0 for
nonzero ~x. Which of these matrices are positive definite?
!
!
!
1 2
1 −2
−1
2
,
,
2 1
−2
6
2 −6
57. (5 points) Let A be a real symmetric 2 × 2 matrix. Prove that the eigenvalues of A
are real numbers.
58. (5 points) Let B be a real 3 × 4 matrix. Prove that the eigenvalues of B T B are
non-negative.
59. (5 points) The spectral theorem says that a symmetric matrix A can be factored into
A = QDQT where Q !is orthogonal and D is diagonal. Find Q and D for the symmetric
3 −1
matrix A =
.
−1 3
60. (5 points) Show that if B is an invertible matrix and A is similar to B, with
A = P BP −1 , then A is invertible.
9
61. (5 points) Write out the singular value decomposition for the matrix A =
2 2
−1 1
!
.
62. (5 points) Strang’s Four Fundamental Subspaces are the nullspace of A, the nullspace
of AT , the row space of A and the column space of A. Describe, using a figure or drawing,
the locations in the matrices U , V of the singular value decomposition A = U ΣV T which
are consumed by the four fundamental subspaces of A.
63. (5 points) Give examples for a vertical shear and a horizontal shear in the plane.
Expected is a 2 × 2 matrix A which represents the linear transformation.
64. (5 points) Give examples for clockwise and counterclockwise rotations in the plane.
Expected is a 2 × 2 matrix A which represents the linear transformation.
65. (5 points) Let the linear transformation T from R3 to R3 be defined by its action on
three independent vectors:
           
5
1
4
0
4
3
           
T 2 = 4 , T 2 = 0 , T 2 = 1 .
1
1
2
1
2
0
Find the unique 3 × 3 matrix A such that T is defined by the matrix multiply equation
T (~x) = A~x.
66. (5 points) Let A be an m × n matrix. Denote by S1 the row space of of A and S2
the column space of A. Prove that T : S1 → S2 defined by T (~x) = A~x is one-to-one and
onto.
Essay Questions
67. (5 points) Define an Elementary Matrix. Display the fundamental matrix multiply
equation which summarizes a sequence of swap, combo, multiply operations, transforming a
matrix A into a matrix B.
68. (5 points) Let V be a vector space and S a subset of V . Define what it means for
S to be a subspace of V . The definition is sometimes called the Subspace Criterion, a
theorem with three requirements, with the conclusion that S is a subspace of V .
69. (5 points) The null space S of an m × n matrix M is a subspace of Rn . This is
10
called the Kernel Theorem, and it is proved from the Subspace Criterion. Both theorems
conclude that some subset is a subspace, but they have different hypotheses. Distinguish
the Kernel theorem from the Subspace Criterion, as viewed from hypotheses.
70. (5 points) Least squares can be used to find the best fit line for the points (1, 2),
(2, 2), (3, 0). Without finding the line equation, describe how to do it, in a few sentences.
71. (5 points) State the Fundamental Theorem of Linear Algebra. Include Part 1: The
dimensions of the four subspaces, and Part 2: The orthogonality equations for the four
subspaces.
72. (5 points) Display the equation for the Singular Value Decomposition (SVD), then
cite the conditions for each matrix. Finish with a written description of how to construct
the matrices in the SVD.
73. (5 points) State the Spectral Theorem for symmetric matrices. Include the important results included in the spectral theorem, about real eigenvalues and diagonalizability.
Then discuss the spectral decomposition.
11